Annals of Mathematics The Brjuno function continuously estimates the size of quadratic Siegel disks By Xavier Buff and Arnaud Ch´eritat Annals of Mathematics, 164 (2006), 265–312 The Brjuno function continuously estimates the size of quadratic Siegel disks By Xavier Buff and Arnaud Ch ´ eritat Abstract If α is an irrational number, Yoccoz defined the Brjuno function Φ by Φ(α)=  n≥0 α 0 α 1 ···α n−1 log 1 α n , where α 0 is the fractional part of α and α n+1 is the fractional part of 1/α n . The numbers α such that Φ(α) < +∞ are called the Brjuno numbers. The quadratic polynomial P α : z → e 2iπα z + z 2 has an indifferent fixed point at the origin. If P α is linearizable, we let r(α) be the conformal radius of the Siegel disk and we set r(α) = 0 otherwise. Yoccoz [Y] proved that Φ(α)=+∞ if and only if r(α) = 0 and that the restriction of α → Φ(α) + log r(α) to the set of Brjuno numbers is bounded from below by a universal constant. In [BC2], we proved that it is also bounded from above by a universal constant. In fact, Marmi, Moussa and Yoccoz [MMY] conjecture that this function extends to R as a H¨older function of exponent 1/2. In this article, we prove that there is a continuous extension to R. Contents 1. Introduction 2. Statement of results 2.1. The value of Υ at rational numbers 2.2. The value of Υ at Cremer numbers 2.3. Strategy of the proof 3. Parabolic explosion 3.1. Outline 3.2. Definitions 3.3. A preliminary lemma: Getting some room for holomorphic motions 3.4. The loss of conformal radius when one removes the exploding cycle 3.5. A short remark: Denominators of convergents and Fibonacci numbers 3.6. The key inequality for the upper bound 3.7. Application to the proof of Theorem 2: Υ at Cremer numbers 266 XAVIER BUFF AND ARNAUD CH ´ ERITAT 4. Proof of inequality (4) (the upper bound) 4.1. Irrational numbers 4.2. Rational numbers: Outline 4.3. Rational numbers 4.4. Proof of Lemma 3: Removing external rays for α close to p/q 5. Yoccoz’s renormalization techniques 5.1. Outline 5.2. Renormalization principle 5.3. Proof of Propostion 3: The uniformization L is close to a linear map 5.4. Controlling the height of renormalization 6. Proof of inequality (5) (the lower bound) in most cases 6.1. Renormalizing a map close to a translation 6.2. Brjuno numbers 6.3. Rational numbers 6.4. Cremer numbers whose P´erez-Marco sum converges 7. Proof of inequality (5) when the P´erez-Marco sum diverges 7.1. Parabolic explosion 7.2. Renormalization Appendix A. Extracts from [BC2] Acknowledgements References 1. Introduction For any irrational number α ∈ R\Q, we denote by (p n /q n ) n≥0 the approx- imants to α given by its continued fraction expansion (by convention, p 0 = α is the integer part of α and q 0 = 1). Remark. Every time we use the notation p/q for a rational number, we mean that q>0 and p and q are coprime. We denote by α∈Z the integer part of α, i.e., the largest integer n ≤ α,by{α} = α −α the fractional part of α, and we define (α n ) n≥0 recursively by setting α 0 = {α} and α n+1 = {1/α n }. We then define β −1 =1 and β n = α 0 α 1 ···α n . Definition 1 (Yoccoz’s Brjuno function). If α is an irrational number, we define Φ(α)= +∞  n=0 β n−1 log 1 α n . THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS 267 If α is a rational number we define Φ(α)=+∞. Irrational numbers for which Φ(α) < ∞ are called Brjuno numbers. Other irrational numbers are called Cremer numbers. Remark. In terms of α n , the definition reads Φ(α) = log 1 α 0 + α 0 log 1 α 1 + α 0 α 1 log 1 α 2 + ··· . Remark. The set B of Brjuno numbers has full measure in R. It con- tains the set of all Diophantine numbers, i.e., numbers for which log q n+1 = O(log q n ). We study the quadratic polynomials P α : z → e 2iπα z + z 2 for α ∈ R. It is known that such P α is linearizable — and so, has a Siegel disk — if and only if α is a Brjuno number. Definition 2. If U Cis a simply connected domain containing 0, we denote by rad(U) the conformal radius of U at 0, i.e., rad(U)=|φ  (0)| where φ :(D, 0) → (U, 0) is any conformal representation. Definition 3. For any Brjuno number α ∈B, we denote by r(α) the confor- mal radius at 0 of the Siegel disk of the quadratic polynomial P α .Ifα ∈ R\B, we define r(α)=0. Remark. The functions α → Φ(α) and α → log r(α), defined on B, are highly discontinuous: for instance they respectively tend to +∞ and −∞ at every rational number. It is known that there exists a constant C 0 such that for any Brjuno number α ∈Band any univalent map f : D → C which fixes 0 with derivative e 2iπα , f has a Siegel disk ∆ f which contains B(0,r) with Φ(α) + log r ≥−C 0 . In particular, for all α ∈B,wehave Φ(α) + log r(α) ≥−C 0 − log 2.(1) Indeed, P α is injective on B(0, 1/2). Remark. The existence of ∆ f is due to Brjuno [Brj]. The lower bound (1) is due to Yoccoz [Y]. In [BC2], we proved that there exists a universal constant C 1 such that for all α ∈B, we have Φ(α) + log r(α) ≤ C 1 .(2) 268 XAVIER BUFF AND ARNAUD CH ´ ERITAT Inequalities (1) and (2) imply that Φ(α) + log r(α) is uniformly bounded on B: (∃C ∈ R), (∀α ∈B), |Φ(α) + log r(α)|≤C.(3) Figure 1: The graph of the function α → Φ(α) + log r(α) with α ∈ [0, 1]. The range is [0, log(2π)]. In this article we prove the following result which was conjectured by Marmi [Ma]. Theorem 1 (Main Theorem). The function α → Φ(α) + log r(α) ex- tends to R as a continuous function. In fact, Marmi, Moussa and Yoccoz made the following stronger conjecture ([MMY] and [Ca]). Conjecture 1. The function α → Φ(α)+log r(α)—which is well-defined on B— is H¨older of exponent 1/2. Remark. Since B is dense in R, being 1/2-H¨older on B and having a 1/2-H¨older extension to R are equivalent, and the extension is unique. Remark. In [Y], Yoccoz uses a modified version of continued fractions. He defines a sequence ˜α n defined by ˜α 0 = d(α, Z) and ˜α n+1 = d(1/˜α n , Z). The corresponding function  Φ defined by  Φ(α)=  n≥0 ˜α 0 ···˜α n−1 log 1 ˜α n has the additional property that  Φ(1 − α)=  Φ(α). Figure 2 shows the graph of the function α →  Φ(α) + log r(α). Theorem 4.6 in [MMY] asserts that the restriction of Φ −  ΦtoB extends to R as a 1/2-H¨older continuous periodic function with period one. It has two consequences: first, the Marmi-Moussa- Yoccoz conjecture is equivalent with Φ replaced by  Φ. Second, with Theorem 1 THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS 269 Figure 2: The graph of the function α →  Φ(α) + log r(α) with α ∈ [0, 1]. The range is [0, log(2π)]. it implies that the function α →  Φ(α) + log r(α) extends to R as a continuous function. 2. Statement of results The function Φ(α) + log r(α) is defined on the set of Brjuno numbers B. In this section, we will define an extension Υ : R → R and in the rest of the article, we will show that for all α ∈ R, lim α  →α, α  ∈B Φ(α  ) + log r(α  )=Υ(α). It is an easy exercise to prove that Υ is then continuous. Remark.Forα ∈ Q, we will give an explicit formula for Υ(α). Definition 4. For α ∈B, we set Υ(α)=Φ(α) + log r(α). 2.1. The value of Υ at rational numbers. A rational number α = p/q ∈ Q has two finite continued fraction expansions, corresponding to two sequences of approximants p n /q n , two sequences α n , and two sequences β n . One of the sequences α n is provided by the usual algorithm: α 0 = {α} and α n+1 = {1/α n }, which eventually gives α m = 0 for some m ∈ N, after which the sequence is not defined any more. The other has the same α k for k<m, its α m = 1, and has one more term, α m+1 =0. 1 In both cases, the sequence β is defined by β −1 = 1 and β n = α 0 ···α n . Let n 0 = m or m + 1 be the last index of the sequence α n of p/q that we 1 A number α  tending to p/q has its α  k that tends to the α k of p/q for all k<m. According to whether α  tends to p/q from the left or the right, α  m tends to one of the two values defined above, that is 0 or 1, the correspondence depending on the parity of m. Moreover, if it is 1, then α  m+1 tends to 0. This motivates the two definitions we made. 270 XAVIER BUFF AND ARNAUD CH ´ ERITAT choose. We have α n 0 = 0. We can form the finite sum Φ trunc (p/q)= n 0 −1  n=0 β n−1 log 1 α n (with the convention that a sum  n=−1 n=0 ··· is equal to 0). It turns out to be independent of the choice between the two values of n 0 , as can easily be checked. Examples. Φ trunc (0/1) = 0 Φ trunc (1/2) = log 2 Φ trunc (1/3) = log 3 Φ trunc (2/3) = log 3 2 + 2 3 log 2 The following two definitions and their relations with the conformal radii of Siegel disks appear in [Ch]. Definition 5. Assume f :(C, 0) → (C, 0) is a germ having a multiple fixed point at the origin whose Taylor expansion is f(z)=z + Az k+1 + O(z k+2 ), with A ∈ C ∗ . The asymptotic size of f at 0 is defined by L a (f,0) =     1 kA     1/k . The map P p/q fixes 0 with derivative e 2iπp/q . Therefore, its q-th iterate is tangent to the identity, and we make the following definition. Definition 6. Assume p/q ∈ Q is a rational number. Then, we define L a (p/q)=L a (P ◦q p/q , 0). For P p/q , it turns out that k = q (see [DH, Ch. IX]). Definition 7. For all rational number p/q, we define Υ  p q  =Φ trunc  p q  + log L a  p q  + log 2π q . THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS 271 Examples (approximate values rounded to the nearest decimal). L a (0/1) = 1 Υ(0/1) = log 2π =1.8379 L a (1/2) = 1 2 Υ(1/2) = log 2π 2 =0.9189 L a (1/3) = 1 3 1 2 7 1 6 Υ(1/3) = log 3 2 − log 7 6 + log 2π 3 =0.8376 L a (2/3) = 1 3 1 2 7 1 6 Υ(2/3) = log 3 2 − log 7 6 + log π 3 =0.6066 . 2.2. The value of Υ at Cremer numbers. Definition 8. For all irrational number α and all integer n ≥ 0, we define Φ n (α)= n  k=0 β k−1 log 1 α k . We recall that a domain U ⊂ C is hyperbolic if and only if its univer- sal cover is isomorphic to D as a Riemann surface. We also recall that it is equivalent to C \ U containing at least two points. Definition 9. If U ⊂ C is a hyperbolic connected domain containing 0, we denote by rad(U) the conformal radius of U at 0, i.e., rad(U)=|π  (0)| where π :(D, 0) → (U, 0) is any universal covering. Remark. This definition of conformal radius coincides with the one given in the introduction in the case of simply connected domains. Definition 10. For all α ∈ R \ Q and all integer n ≥ 0, we define X n (α)={z ∈ C ∗ | z is a periodic point of P α of period ≤ q n } where p n /q n are the approximants to α, r n (α) = rad(C \ X n (α)) and d n (α)=d(0,X n (α)). Remark.Ifn ≥ 2, then q n ≥ 2, X n (α) contains at least two points and r n (α) ∈]0,+∞[ . Moreover, for n ≥ 2, the function α → log r n (α) is well- defined and continuous in a neighborhood of every point α ∈ R \ Q. For all irrational number α, the sequence (r n (α)) n≥0 is decreasing and converges to r(α)asn →∞. Indeed, if 0 is not linearizable, it is accumulated by periodic points of P α . 2 If 0 is linearizable, the Siegel disk ∆ α is contained in C \X n (α) for all n ≥ 0 and the boundary of ∆ α is accumulated by periodic 2 In fact, Yoccoz proved that 0 is accumulated by whole cycles. 272 XAVIER BUFF AND ARNAUD CH ´ ERITAT points of P α . 3 Since P α is tangent the rotation of angle α and α is irrational, if 0 is not linearizable, then r n (α) ∼ n→+∞ d n (α). If α is a Brjuno number, then lim n→∞ Φ n (α) + log r n (α)=Υ(α). In Section 3, we will prove the following theorem. Theorem 2. For all Cremer numbers α, the sequence Φ n (α) + log r n (α) has a finite limit when n −→ +∞. Definition 11. For all Cremer numbers α, we define Υ(α) = lim n→+∞ Φ n (α) + log r n (α). Remark. This definition is equivalent to Υ(α) = lim n→+∞ Φ n (α) + log d n (α). 2.3. Strategy of the proof. Our goal is to prove that for all α ∈ R, the value of Υ(α) defined previously (see Definitions 4, 7 and 11) is the limit of Φ(α  ) + log r(α  )asα  ∈Btends to α. The strategy consists in bounding Φ(α  ) + log r(α  ) from above and from below as α  ∈Btends to α. The upper bound follows from techniques of parabolic explosion developed in [Ch] and [BC2]. We present them in Section 3, and in Section 4 we show that for all α ∈ R, lim sup α  →α, α  ∈B Φ(α  ) + log r(α  ) ≤ Υ(α).(4) The lower bound essentially follows from techniques of renormalization introduced by Yoccoz in [Y]. He uses estimates which are valid for all maps which are univalent in D and fix 0 with derivative of modulus 1. In our case, we will need to improve those estimates for maps which are close to rotations and maps which have at most one fixed point in D ∗ (see §5). In Sections 6 and 7 we show that for all α ∈ R, lim inf α  →α, α  ∈B Φ(α  ) + log r(α  ) ≥ Υ(α).(5) Let us mention that inequality (4) without inequality (5) (respectively inequality (5) without inequality (4)) is not sufficient to conclude that Υ is upper semi-continuous (respectively lower semi-continuous) since we only con- sider approximating α by sequences of Brjuno numbers. 3 It is not known whether ∂∆ α is always accumulated by whole cycles. THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS 273 3. Parabolic explosion In this section, we first present the techniques of parabolic explosion. We then apply those techniques in order to prove Theorem 2. 3.1. Outline. Here, we informally describe what will be done in Section 3. Let α be irrational. Recall that r n−1 (α) is the conformal radius at 0 of the complement of X n−1 (α), the set of nonzero periodic points of period ≤ q n−1 . When we increment n −1ton, X n−1 (α) contains more periodic points, hence r n−1 (α) decreases. Among the points removed from C \ X n−1 (α), we single out a particular cycle C. We will prove that this cycle induces a decrease in conformal radius, of at least β n−1 log 1 α n , up to a tame error term. What is this cycle C? The approximant p n /q n is close to α. Therefore P α is a perturbation of P p n /q n . The latter has a parabolic fixed point at 0. The perturbations of P p n /q n have a cycle C of period q n close to 0. Why a decrease of β n−1 log 1 α n ? The points in the cycle turn out to de- pend analytically on the q n -th root of the perturbation. It follows from a ver- sion of Schwarz’s lemma that the cycle cannot go significantly farther than |α − p n /q n | 1/q n times the conformal radius of the region where the explo- sion takes place. We will see that the cycle cannot collide with the points of X n−1 (α). In terms of logarithms of conformal radii, this implies that there must be a decrease of −1 q n log |α − p n /q n |. The theory of continued fractions approximates this value by β n−1 log 1 α n . Unfortunately there are several technical difficulties. They will induce error terms of order 1 q n log q n . Among them: • One needs p n /q n to be a good enough approximant to α. When it is not, the claimed decrease may not be true, but it is then small enough to be swallowed by the error term. • The set X n−1 (α) depends on α and thus, during the explosion, the cycle avoids a set which moves with α. We have to show that this motion is small (by proving that there is a holomorphic motion defined on a domain in the parameter space much bigger than the domain on which the explosion is defined). And we have to prove that this small motion induces a small error term. Other technical difficulties are addressed in this section. 3.2. Definitions. Assume p/q ∈ Q is a rational number. The origin is a parabolic fixed point for the quadratic polynomial P p/q . It is known (see [DH, Ch. IX]) that there exists a complex number A ∈ C ∗ such that P ◦q p/q (z)=z + Az q+1 + O(z q+2 ). Thus, P ◦q p/q has a fixed point of multiplicity q + 1 at the origin. By Rouch´e’s theorem, when α is close to p/q, the polynomial P ◦q α has q + 1 fixed points [...]... explain how we choose ρn0 On the one hand, it follows from our techniques of parabolic explosion that the distance dn0 (α ) 291 THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS between 0 and Cpn0 /qn0 (α ) satisfies log dn0 (α ) ≤ −Φn0 (α ) + C0 On the other hand, it follows from Yoccoz’s lower bound on the size of Siegel disks and from parabolic explosion that the other cycles of period ≤ qn0 lie outside... (θ), θ ∈ Θ, form a cycle of rays which land on the cycle Cp/q (α ) We denote by Y (α ) the union of Cp/q (α ) and this cycle of rays Figure 3 shows the rays of argument 1/7, 2/7 √ 4/7 and the boundary of and √ the Siegel disk for the polynomial P(1/3)+ε for ε = 2/1000 and ε = 2/10000 Figure 3: The rays of argument 1/7, 2/7 and 4/7 and the boundary of the √ Siegel disk for the polynomial P(1/3)+ε :... Vn (α ) = Vn−1 (α ) otherwise Then, the hypotheses of Lemma 2 are satisfied and (as in Proposition 2), we have log rad(Vn (α )) − log rad(Vn−1 (α )) ≤ log |α − pn /qn | log qn +C qn qn ≤ −Φn (α ) + Φn−1 (α ) + (C − 1) log qn , qn 283 THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS where C is the constant provided by Lemma 2 The Siegel disk ∆α is contained in the intersection of the sets Vn (α ), and... periodic cycle of rays which land at 0 281 THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS If p/q ∈ Z, the dynamical ray of argument 0 is fixed and lands at 0 We set θ− = θ+ = 0 and Θ = {0} Let us recall the following rule: the ray Rα (θ) moves holomorphically with α as long as c does not belong to the closure of the union of the RM (2k θ) for k ∈ N∗ Definition 13 When α ∈ R is close to p/q, the rays Rα... 293 THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS ∼α 1 ∼1/α 1 F (l) l L W U Z F (Z ) F ◦k (Z ) F (0) L(Z) L(Z)+1 L(Z)+k l 0 0 1 Figure 6: Construction of the map L : W → H Proposition 3, for all z ∈ H, if Im(Z) > 6δ/αn , then there exists an integer k such that Z − k belongs to D, the domain of definition of Hn Then, D+Z contains the half-plane “Im(Z) > 6δ/αn ” Moreover, the map Hn commutes with the. .. 4πα α 5.4 Controlling the height of renormalization In this section, we determine an upper bound for the height t above which the fundamental estimates (6) are satisfied The first result is due to Yoccoz (it easily follows from the compactness of S(0) , but the interested reader can find sharper bounds in [Y], in the lemma of §3.2, p 26) 299 THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS Proposition 8... ) + O(ε1+1/q w2 ) + O(ε2 w) THE BRJUNO FUNCTION AND THE SIZE OF SIEGEL DISKS 285 Figure 4 shows some trajectories of the real flow of the vector field ∂ 2iπqw(1 − wq ) ∂w for q = 3 The origin is a center and its basin Ω is colored light grey ∂ Figure 4: Some trajectories of the real flow of the vector field 2iπqw(1 − wq ) ∂w for q = 3 Let us now define Yε = 1 Y zε p +ε q The set Yε contains 1 and we have... tends to θ When α is real, the parameter c is on the boundary of the main cardioid of the Mandelbrot set If α = p/q ∈ Z, then c = 1/4 and there are two / external rays of M landing at c We denote by θ− < θ+ their arguments in ]0, 1[ The arguments θ+ and θ− are periodic of period q under multiplication by 2 modulo 1 They belong to the same orbit Θ In the dynamical plane of Pp/q , the rays Rp/q (θ), θ ∈... a Siegel disk Assume α is a Brjuno number Let φ : D → ∆(α) be the linearization Conjugating by φ−1 , the family Pα becomes a family gα of maps tending to Rα , uniformly on every compact subset of D, as α −→ α We will give a lower bound on the size the Siegel disks ∆(gα ) Conjugating back by φ multiplies conformal radii by r(α), and the Siegel disk of Pα must contain φ(∆(gα )) Consider the sequence of. .. In other words, the points of the cycles depend analytically, not on the perturbation α − p/q but on its q-th root δ Moreover, these q points are given by a single analytic function χ, applied to the q values of the q-th root The proposition also gives a lower bound on the size of the disk on which this holds Remark B(p/q, 1/q 3 ) Observe that δ ∈ B(0, 1/q 3/q ) if and only if α = p/q + δ q ∈ In the . Annals of Mathematics The Brjuno function continuously estimates the size of quadratic Siegel disks By Xavier Buff and. Arnaud Ch´eritat Annals of Mathematics, 164 (2006), 265–312 The Brjuno function continuously estimates the size of quadratic Siegel disks By Xavier Buff and